Any line bundle (invertible sheaf) on $\operatorname{Spec}k$ is trivial?

Question

Literally, let $\mathcal{L}$ be a invertible module on $\operatorname{Spec}k$. Then $\mathcal{L}$ is trivial?

Here what the 'triviality' means?

It seems to be clear from the definition of invertible module (locally free module of rank 1), but it is certainly true?

Answer 1

A line bundle $\mathcal{L}$ on a scheme (or locally ringed space) $X$ is called trivial if $\mathcal{L}\cong\mathcal{O}_X$ as $\mathcal{O}_X$-modules.

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As $\operatorname{Spec} k$ is a one-point space, locally free means free: every point has an open neighborhood where $\mathcal{L}\cong\mathcal{O}_X$. But there's only one point and only one non-empty open set, so there you go.